Vertical sextant angle nails lighthouse height

Being captain and frequent instructor of marine navigation aboard the training vessel Ocean Star, I find, on occasion, circumstances where a knowledge of basic plane trigonometry can be useful.

We have found on one of our many cruises to the Dry Tortugas (not far from Key West, Fla.), that just finding our distance off by using a vertical sextant angle is only half the fun!

On Garden Key in the Dry Tortugas there exists an abandoned lighthouse the height of which is not listed on charts. Even though this abandoned lighthouse has an actual light, the light itself cannot officially be used for normal navigational purposes. The light tower is built atop a fortress wall which is part of historic Fort Jefferson. Its position behind the southeast bastion on the chart is correct, however, which does allow us to use it for visual bearings. Since our little trigonometric exercise, we now can use this abandoned lighthouse to determine our distance off as well.

On a trip last winter, the first mate and I decided we could determine the height of the light by vertical sextant angle by reversing the procedure we normally would use to determine distance off. We then compared our calculated height-to-height figures off blueprints that we obtained from Wayne Landrum of the National Park Service. It was astonishing and inspiring to discover that our figures for the height of the light and its lighthouse were all within two to three inches!

We measured the lighthouse two ways: First, we took a vertical sextant angle with my Cassens and Plath sextant, measuring from the shoreline (water's edge) to the lens house of the light itself, which was an angle of 2° 47.2', giving us an overall height of 62.246 feet. We then compared this to the blueprints of the structure to the light above the wall.

Plans showed the height of the lens above the wall to be 31 feet. Then we measured by tape measure the distance from the top of the wall to the base of the wall, which came to 31.4375 feet, or 31 feet, 5 1/4 inches. Add these two together and the overall light height comes out to 62.4375 feet. This is within inches of our calculations for the light.

Now, we decided we wanted to take this further. We then took a vertical sextant angle from the shoreline (water's edge) to the highest point of the lighthouse itself. That angle was 3° 00.6', giving us a height of 67.229 feet. We did find in the U.S. Coast Pilot #5 the stated fact that the height of the lighthouse is 67 feet. Again, the two figures are within inches of each other, making me a believer in vertical sextant angles!

Now, not only can I use the charted lighthouse for visual bearings, but I can rely on the height of the lighthouse to give me the distance off as well. The end result is, and can be, a reliable fix!

As for the particulars, we needed to know our distance off at the time of taking these angles to determine the other side of the triangle (the height of the light). We determined this by use of a new tool on the market that we happened to have aboard. It's called a Riegl Lasertape FG21, and it employs a pulsed infrared laser that has a maximum range of 1.9 nm or 3,850 yards, with half-yard accuracy. The Lasertape works in a similar fashion to a radar: Pulses are emitted from the unit, and some of the laser energy is bounced off the target and returned. The unit determines distance by noting the elapsed time between transmission of the pulse train and a return. This time is divided by two and then multiplied by the speed of light.

The distance we measured for our problem was 0.15 nm. This worked out to be 425 yards or 1,275 feet. The index error on the Cassens and Plath sextant was 0.5' off the arc.

Virginia Wagner is staff captain of the 88-foot steel schooner Ocean Star, which conducts navigational training trips for this magazine.